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The Ideal Gas Law Plus, the density formula and Avogadros Law

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Review Gases are made of particles that move rapidly and randomly. Gases are made of particles that move rapidly and randomly. Temperature is a measure of how rapidly the molecules in a gas are moving on average. Temperature is a measure of how rapidly the molecules in a gas are moving on average. Collisions between gas molecules surface of an object (or the walls of a container) give rise to gas pressure. Collisions between gas molecules surface of an object (or the walls of a container) give rise to gas pressure. Standard temperature and pressure Standard temperature and pressure 273 K (0ºC) 273 K (0ºC) kPa (1 atm) kPa (1 atm)

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Ideal Gases An ideal gas is one that perfectly obeys the predictions made by the KMT: An ideal gas is one that perfectly obeys the predictions made by the KMT: Its molecules have zero diameter. Its molecules have zero diameter. Its molecules have zero intermolecular forces. Its molecules have zero intermolecular forces. Collisions are always elastic. Collisions are always elastic. Gases arent perfectly ideal. Gases arent perfectly ideal. Some gases approach idealness under certain conditions: Some gases approach idealness under certain conditions: high temperature high temperature low pressure low pressure

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Ideal vs. Real Gases The boiling point of nitrogen (N 2 ) is Kelvins ( ºC). The boiling point of nitrogen (N 2 ) is Kelvins ( ºC). At room temperature (298 K) and standard pressure (1 atm), nitrogen behaves very much like an ideal gas. At room temperature (298 K) and standard pressure (1 atm), nitrogen behaves very much like an ideal gas. Room temperature is far above nitrogens boilling point. Room temperature is far above nitrogens boilling point. At 78 K and 30 atm, nitrogens behavior isnt so ideal. At 78 K and 30 atm, nitrogens behavior isnt so ideal. Its molecules are just barely moving fast enough to remain in the gas state. Its molecules are just barely moving fast enough to remain in the gas state. Intermolecular forces affect the behavior of N 2 at such a low temperature. Intermolecular forces affect the behavior of N 2 at such a low temperature. High pressures squeeze the molecules close together, increasing the effects of their intermolecular forces. High pressures squeeze the molecules close together, increasing the effects of their intermolecular forces.

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Ideal vs. Real Gases Ideal Gas

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Ideal vs. Real Gases In a gas that approximates ideal behavior, molecules are fast-moving and far apart. At very low temperatures, gas molecules become sluggish and attractive forces alter their behavior. At very high pressures gas molecules are squeezed close together and molecular interactions become far more common (and important).

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The Ideal Gas Law PV = nRT PV = nRT P = pressure (atm) P = pressure (atm) V = volume (L) V = volume (L) n = moles n = moles R = Universal Gas Constant R = Universal Gas Constant R = L*atm / mol*K R = L*atm / mol*K T = temperature (K) T = temperature (K) Applies to gases that exhibit ideal behavior. Applies to gases that exhibit ideal behavior. For non-ideal gases (gases at very low temperature or extremely high pressures) PV nRT. For non-ideal gases (gases at very low temperature or extremely high pressures) PV nRT.

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The Ideal Gas Law In order to use R = L*atm / mol*K, In order to use R = L*atm / mol*K, Pressure must be expressed in atm. Pressure must be expressed in atm. Volume must be expressed in L. Volume must be expressed in L. Temperature must be expressed in K. Temperature must be expressed in K. You should get used to converting between different units of pressure, volume, and temperature. You should get used to converting between different units of pressure, volume, and temperature. There are other values of R for use with other units, but instead of learning many different values for R, you should learn how to convert. There are other values of R for use with other units, but instead of learning many different values for R, you should learn how to convert.

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The Ideal Gas Law What is the volume of 3.00 moles of helium at a temperature of 400. K and a pressure of 1.50 atm? What is the volume of 3.00 moles of helium at a temperature of 400. K and a pressure of 1.50 atm? PV = nRT PV = nRT (1.50 atm) V = (3.00 mol)( L*atm / mol*K ) (400. K) (1.50 atm) V = (3.00 mol)( L*atm / mol*K ) (400. K) (1.50 atm) V = 98.5 L*atm (1.50 atm) V = 98.5 L*atm V = 65.7 L V = 65.7 L

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The Ideal Gas Law What is the pressure exerted by moles of nitrogen gas in a 45.0 L container if the temperature is 350.ºC? What is the pressure exerted by moles of nitrogen gas in a 45.0 L container if the temperature is 350.ºC? First we must convert ºC to K? First we must convert ºC to K? 350.ºC = 623 K 350.ºC = 623 K PV = nRT PV = nRT P(45.0 L) = (0.400 mol)( L*atm / mol*K )(623 K) P(45.0 L) = (0.400 mol)( L*atm / mol*K )(623 K) P(45.0 L) = 20.5 L*atm P(45.0 L) = 20.5 L*atm P = atm P = atm

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The Ideal Gas Law At what temperature (in ºC) will 2.00 grams of argon gas exert a pressure of 12.5 atm in a 160.-mL container? At what temperature (in ºC) will 2.00 grams of argon gas exert a pressure of 12.5 atm in a 160.-mL container? First we must convert from grams of Ar to moles: First we must convert from grams of Ar to moles: 2.00 g Ar x = mol Ar 2.00 g Ar x = mol Ar Next, convert from mL to L: Next, convert from mL to L: 160. mL x = L 160. mL x = L g Ar 1 mol Ar 1000 mL 1 L

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The Ideal Gas Law At what temperature (in ºC) will mol of argon gas exert a pressure of 12.5 atm in a L container? At what temperature (in ºC) will mol of argon gas exert a pressure of 12.5 atm in a L container? PV = nRT PV = nRT (12.5 atm)(0.160 L) = ( mol)( L*atm / mol*K ) T (12.5 atm)(0.160 L) = ( mol)( L*atm / mol*K ) T 2.00 atm*L = ( L*atm/K) T 2.00 atm*L = ( L*atm/K) T T = 486 K T = 486 K T = 213º C T = 213º C

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Gas Density Density = mass / volume Density = mass / volume The volume of an ideal gas is: The volume of an ideal gas is: V = nRT / P V = nRT / P Putting the two equations together: Putting the two equations together: D = mP / nRT D = mP / nRT The molar mass of a gas is the number of grams (m) per mole (n), so: The molar mass of a gas is the number of grams (m) per mole (n), so: Molar mass (M) = m/n Molar mass (M) = m/n For an ideal gas, D = PM / RT For an ideal gas, D = PM / RT

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Gas Density What is the density of chlorine gas, Cl 2, at atm and 300. K? What is the density of chlorine gas, Cl 2, at atm and 300. K? The molar mass of Cl 2 is: The molar mass of Cl 2 is: 2 x Cl = 2 x g/mol = g/mol 2 x Cl = 2 x g/mol = g/mol D = PM / RT D = PM / RT D = (0.950 atm)( g/mol) / ( L*atm / mol*K )(300. K) D = (0.950 atm)( g/mol) / ( L*atm / mol*K )(300. K) D = 2.73 g/L D = 2.73 g/L 1 Liter of Cl 2 under these conditions would weigh 2.73 grams. 1 Liter of Cl 2 under these conditions would weigh 2.73 grams.

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Gas Density What is the density of CO 2 gas at STP? What is the density of CO 2 gas at STP? The molar mass of CO 2 is: The molar mass of CO 2 is: 1 x C = 1 x g/mol = g/mol 1 x C = 1 x g/mol = g/mol 2 x O = 2 x g/mol = g/mol 2 x O = 2 x g/mol = g/mol Total = g/mol Total = g/mol Standard temperature = 273 K Standard temperature = 273 K Standard pressure = 1.00 atm Standard pressure = 1.00 atm D = PM / RT D = PM / RT D = (1.00 atm)( g/mol) / ( L*atm / mol*K )(273 K) D = (1.00 atm)( g/mol) / ( L*atm / mol*K )(273 K) D = 1.96 g/L D = 1.96 g/L

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Avogadros Law In addition to having a very large number named after him, Amedeo Avogadro made a very important deduction about gases. In addition to having a very large number named after him, Amedeo Avogadro made a very important deduction about gases. Equal volumes of ideal gases at equal temperatures and pressures contain equal numbers of molecules. Equal volumes of ideal gases at equal temperatures and pressures contain equal numbers of molecules. Doesnt matter what the identity of the gas is. Doesnt matter what the identity of the gas is. In other words, if you have two different ideal gases under identical conditions, the molar volume of the gases is the same. In other words, if you have two different ideal gases under identical conditions, the molar volume of the gases is the same. At STP, 1 mole of any gas = 22.4 L At STP, 1 mole of any gas = 22.4 L

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Avogadros Law What is the volume of 2.00 moles of O 2 at STP? (Assume ideal behavior.) What is the volume of 2.00 moles of O 2 at STP? (Assume ideal behavior.) 1 mol of any STP = 22.4 L 1 mol of any STP = 22.4 L 2.00 mol O 2 x = 44.8 L O mol O 2 x = 44.8 L O 2 1 mol O L O 2

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Avogadros Law

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A Reminder Gas law problems dont always have to come with easy-to-use units. Gas law problems dont always have to come with easy-to-use units. They often require unit conversions before they can be solved. They often require unit conversions before they can be solved. You should always ask yourself What does this problem want me to find, and what am I given? You should always ask yourself What does this problem want me to find, and what am I given? Make a plan to solve the problem based on the available information. Make a plan to solve the problem based on the available information. Not every problem will be the same. Not every problem will be the same. If you approach all gas law problems expecting them to be identical, you will be frustrated. If you approach all gas law problems expecting them to be identical, you will be frustrated. Dont get lazy or sloppy in your work! Dont get lazy or sloppy in your work!

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